Boundary-Nonregular Functions in the Disc Algebra and in Holomorphic Lipschitz Spaces

We prove in this paper the existence of dense linear subspaces in the classical holomorphic Lipschitz spaces in the disc all of whose non-null functions are nowhere differentiable at the boundary. Infinitely generated free algebras as well as infinite dimensional Banach spaces consisting of Lipschit...

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Bibliographic Details
Published in:Mediterranean journal of mathematics Vol. 15; no. 3; pp. 1 - 20
Main Authors: Bernal-González, L., Bonilla, A., López-Salazar, J., Seoane-Sepúlveda, J. B.
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.06.2018
Springer Nature B.V
Subjects:
Algebra
Banach spaces
Borel sets
Mathematical analysis
Mathematics
Mathematics and Statistics
Subspaces
lipschitzian function
Lineability
Secondary 15A03
46E10
Nowhere differentiable function
Disc algebra
26A16
26A27
Spaceability
Primary 30H50
Algebrability
ISSN:1660-5446, 1660-5454
Online Access:Get full text
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Summary:We prove in this paper the existence of dense linear subspaces in the classical holomorphic Lipschitz spaces in the disc all of whose non-null functions are nowhere differentiable at the boundary. Infinitely generated free algebras as well as infinite dimensional Banach spaces consisting of Lipschitz functions enjoying the mentioned property almost everywhere on the boundary are also exhibited. It is also investigated the algebraic size of the family of functions in the disc algebra that either do not preserve Borel sets on the unit circle or possess the Cantor boundary behavior on the disc.
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ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-018-1160-6